Question: Paige launched a ball using a catapult she built. The height of the ball (in meters above the ground) $t$ seconds after launch is modeled by $h(t)=-5t^2+40t$ Paige wants to know when the ball will hit the ground. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. $h(t)=$ 2) How many seconds after launch does the ball hit the ground?
Answer: Choosing a form When the ball hits the ground, its height above the ground is $0$ meters. So we're looking for what values of $t$ make the output of the function $0$. Which form reveals this feature? Here's a summary of what each form reveals along with examples. Note that these are all equivalent forms of the same function, but not the function modeling the height of the ball. Form Example Feature revealed Standard $f(x)=2x^2-12x+{10}$ $y$ -intercept is ${10}$ Factored $f(x)=2(x-C{1})(x-C{5})$ Zeros are $x=C1$ and $x=C5$ Vertex $f(x)=2(x-{3})^2{-8}$ Vertex is $(3,{-8})$ Rewrite in factored form The zeros of the function tell us which values of $t$ make the output of the function $0$, so let's rewrite $h(t)$ in factored form: $\begin{aligned} h(t)&=-5t^2+40t \\\\ &=-5t(t-8) \end{aligned}$ When does the ball hit the ground? The factored form of the function reveals its zeros: $\begin{aligned} &0=-5t(t-8) \\\\ &-5t=0\text{ or }t-8=0 \\\\ &\xcancel{t=0} \text{ or }t=8 \end{aligned}$ The ball launches at $t=0$, so the ball hits the ground at $t=8$ seconds. Answers 1) The factored form of the function reveals when the ball hits the ground: $h(t)=-5t(t-8)$ 2) The ball hits the ground $8$ seconds after launch.